Dot Product Commutativity

If I had an equation that said

del dot u = 0 (1)

And then I had another equation that used a term

(u dot del) u

Then wouldn't that term just equal 0?

http://pages.cs.wisc.edu/~chaol/data/cs777/stam-stable_fluids.pdf

I'm trying to understand this here paper, and the Advection term in the Navier-Stokes Equation.

http://en.wikipedia.org/wiki/Advection#The_advection_equation

But why is it that even though it says del dot u is 0, the u dot del term is not set to 0?  Are vector dot products not commutative?

The operators are not associative nor commutative. This means both the order of the operators and how you bracket them matters.

It's worth studying this issue with some care, as it is rather fundamental.